Abstract
We consider a class of systems of nonlinear ordinary differential equations with parameters. In particular, systems of such type arise when modeling the multistage synthesis of a substance. We study properties of solutions to the systems and propose a method for approximate solving the systems in the case of very large coefficients. We establish approximation estimates and show that the convergence rate depends on the parameters characterizing the nonlinearity of the systems. Moreover, the larger the coefficients of the systems, the more exact the approximate solutions. Thereby this method allows us to avoid difficulties arising inevitably when solving systems of nonlinear differential equations with very large coefficients.
I. Matveeva—The work is supported in part by the Russian Foundation for Basic Research (project no. 16-01-00592).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Goodwin, B.C.: Oscillatory behavior of enzymatic control processes. Adv. Enzyme Reg. 3, 425–439 (1965)
Tyson, J.J., Othmer, H.G.: The dynamics of feedback control circuits in biochemical pathways. Prog. Theor. Biol. 5, 1–62 (1978)
Smolen, P., Baxter, D.A., Byrne, J.H.: Modeling transcriptional control in gene networks – methods, recent results, and future directions. Bull. Math. Biol. 62, 247–292 (2000)
de Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9, 69–103 (2002)
Likhoshvai, V.A., Fadeev, S.I., Demidenko, G.V., Matushkin, Y.G.: Modeling multistage synthesis without branching by a delay equation. Sib. Zh. Ind. Mat. 7, 73–94 (2004)
Demidenko, G.V., Likhoshvai, V.A.: On differential equations with retarded argument. Sib. Math. J. 46, 417–430 (2005)
Demidenko, G.V., Khropova, Y.E.: Matrix process modelling: on properties of solutions of one delay differential equation. In: Proceedings of the Fifth International Conference on Bioinformatics of Genome Regulation and Structure, vol. 3, pp. 38–42. Institute of Cytology and Genetics, Novosibirsk (2006)
Demidenko, G.V., Likhoshvai, V.A., Kotova, T.V., Khropova, Y.E.: On one class of systems of differential equations and on retarded equations. Sib. Math. J. 47, 45–54 (2006)
Demidenko, G.V., Likhoshvai, V.A., Mudrov, A.V.: On the relationship between solutions of delay differential equations and infinite-dimensional systems of differential equations. Diff. Equ. 45, 33–45 (2009)
Demidenko, G.V., Mel’nik, I.A.: On a method of approximation of solutions to delay differential equations. Sib. Math. J. 51, 419–434 (2010)
Demidenko, G.V., Kotova, T.V.: Limit properties of solutions to one class of systems of differential equations with parameters. J. Anal. Appl. 8, 63–74 (2010)
Demidenko, G.V.: Systems of differential equations of higher dimension and delay equations. Sib. Math. J. 53, 1021–1028 (2012)
Demidenko, G.V.: On classes of systems of differential equations of large dimensions and delay equations. Math. Forum 5, 45–56 (2011). (Itogi Nauki. Yug Rossii)
Kotova, T.V., Mel’nik, I.A.: On properties of solutions of one nonlinear system to differential equations with parameters. Preprint/Sobolev Inst. Mat. 253, 17 (2010). Novosibirsk
Matveeva, I.I., Mel’nik, I.A.: On the properties of solutions to a class of nonlinear systems of differential equations of large dimension. Sib. Math. J. 53, 248–258 (2012)
Uvarova, I.A.: On a system of nonlinear differential equations of high dimension. J. Appl. Ind. Math. 8, 594–603 (2014)
Krasovskii, N.N.: The approximation of a problem of analytic design of controls in a system with time-lag. J. Appl. Math. Mech. 28, 876–885 (1964)
Repin, Y.M.: On the approximate replacement of systems with lag by ordinary dynamical systems. J. Appl. Math. Mech. 29, 254–264 (1965)
Banks, H.T., Burns, J.A.: Hereditary control problems: numerical methods based on averaging approximations. SIAM J. Control Optim. 16, 169–208 (1978)
Györi, I.: Two approximation techniques for functional differential equations. Comput. Math. Appl. 16, 195–214 (1988)
Kraszanai, B., Györi, I., Pituk, M.: The modified chain method for a class of delay differential equations arising in neural networks. Math. Comput. Model. 51, 452–460 (2010)
Ilika, S.A., Cherevko, I.M.: Approximation of nonlinear differential functional equations. J. Math. Sci. 190, 669–682 (2013)
Matveeva, I.I., Popov, A.M.: Matrix process modelling: dependence of solutions of a system of differential equations on parameter. In: Proceedings of the Fifth International Conference on Bioinformatics of Genome Regulation and Structure, vol. 3, pp. 82–85. Institute of Cytology and Genetics, Novosibirsk (2006)
Matveeva, I.I.: On properties of solutions to a system of differential equations with a parameter. J. Anal. Appl. 7, 75–84 (2009)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Vol. 3: Theory of Differential Equations. Academic Press, New York, London (1967)
Hartman, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Matveeva, I. (2017). On Approximate Solutions to One Class of Nonlinear Differential Equations. In: Giri, D., Mohapatra, R., Begehr, H., Obaidat, M. (eds) Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655. Springer, Singapore. https://doi.org/10.1007/978-981-10-4642-1_19
Download citation
DOI: https://doi.org/10.1007/978-981-10-4642-1_19
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-4641-4
Online ISBN: 978-981-10-4642-1
eBook Packages: Computer ScienceComputer Science (R0)